Combinatorics: the Art of Counting

Total Page:16

File Type:pdf, Size:1020Kb

Combinatorics: the Art of Counting GRADUATE STUDIES IN MATHEMATICS 210 Combinatorics: The Art of Counting Bruce E. Sagan Combinatorics: The Art of Counting GRADUATE STUDIES IN MATHEMATICS 210 Combinatorics: The Art of Counting Bruce E. Sagan Marco Gualtieri Bjorn Poonen Gigliola Staffilani (Chair) Jeff A. Viaclovsky 2020 Mathematics Subject Classification. Primary 05-01; Secondary 06-01. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-210 Library of Congress Cataloging-in-Publication Data Names: Sagan, Bruce Eli, 1954- author. Title: Combinatorics : the art of counting / Bruce E. Sagan. Description: Providence, Rhode Island : American Mathematical Society, [2020] | Series: Gradu- ate studies in mathematics, 1065-7339 ; 210 | Includes bibliographical references and index. Identifiers: LCCN 2020025345 | ISBN 9781470460327 (paperback) | ISBN 9781470462802 (ebook) Subjects: LCSH: Combinatorial analysis–Textbooks. | AMS: Combinatorics – Instructional ex- position (textbooks, tutorial papers, etc.). | Order, lattices, ordered algebraic structures – Instructional exposition (textbooks, tutorial papers, etc.). Classification: LCC QA164 .S24 2020 | DDC 511/.6–dc23 LC record available at https://lccn.loc.gov/2020025345 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2020 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10987654321 252423222120 To Sally, for her love and support Contents Preface xi List of Notation xiii Chapter 1. Basic Counting 1 §1.1. The Sum and Product Rules for sets 1 §1.2. Permutations and words 4 §1.3. Combinations and subsets 5 §1.4. Set partitions 10 §1.5. Permutations by cycle structure 11 §1.6. Integer partitions 13 §1.7. Compositions 16 §1.8. The twelvefold way 17 §1.9. Graphs and digraphs 18 §1.10. Trees 22 §1.11. Lattice paths 25 §1.12. Pattern avoidance 28 Exercises 33 Chapter 2. Counting with Signs 41 §2.1. The Principle of Inclusion and Exclusion 41 §2.2. Sign-reversing involutions 44 §2.3. The Garsia–Milne Involution Principle 49 §2.4. The Reflection Principle 52 vii viii Contents §2.5. The Lindström–Gessel–Viennot Lemma 55 §2.6. The Matrix-Tree Theorem 59 Exercises 64 Chapter 3. Counting with Ordinary Generating Functions 71 §3.1. Generating polynomials 71 §3.2. Statistics and 푞-analogues 74 §3.3. The algebra of formal power series 81 §3.4. The Sum and Product Rules for ogfs 86 §3.5. Revisiting integer partitions 89 §3.6. Recurrence relations and generating functions 92 §3.7. Rational generating functions and linear recursions 96 §3.8. Chromatic polynomials 99 §3.9. Combinatorial reciprocity 106 Exercises 109 Chapter 4. Counting with Exponential Generating Functions 117 §4.1. First examples 117 §4.2. Generating functions for Eulerian polynomials 121 §4.3. Labeled structures 124 §4.4. The Sum and Product Rules for egfs 128 §4.5. The Exponential Formula 131 Exercises 134 Chapter 5. Counting with Partially Ordered Sets 139 §5.1. Basic properties of partially ordered sets 139 §5.2. Chains, antichains, and operations on posets 145 §5.3. Lattices 148 §5.4. The Möbius function of a poset 154 §5.5. The Möbius Inversion Theorem 157 §5.6. Characteristic polynomials 164 §5.7. Quotients of posets 168 §5.8. Computing the Möbius function 174 §5.9. Binomial posets 178 Exercises 183 Chapter 6. Counting with Group Actions 189 §6.1. Groups acting on sets 189 §6.2. Burnside’s Lemma 192 §6.3. The cycle index 197 Contents ix §6.4. Redfield–Pólya theory 200 §6.5. An application to proving congruences 205 §6.6. The cyclic sieving phenomenon 209 Exercises 213 Chapter 7. Counting with Symmetric Functions 219 §7.1. The algebra of symmetric functions, Sym 219 §7.2. The Schur basis of Sym 224 §7.3. Hooklengths 230 §7.4. 푃-partitions 235 §7.5. The Robinson–Schensted–Knuth correspondence 240 §7.6. Longest increasing and decreasing subsequences 244 §7.7. Differential posets 248 §7.8. The chromatic symmetric function 253 §7.9. Cyclic sieving redux 256 Exercises 259 Chapter 8. Counting with Quasisymmetric Functions 267 §8.1. The algebra of quasisymmetric functions, QSym 267 §8.2. Reverse 푃-partitions 270 §8.3. Chain enumeration in posets 274 §8.4. Pattern avoidance and quasisymmetric functions 276 §8.5. The chromatic quasisymmetric function 279 Exercises 283 Appendix. Introduction to Representation Theory 287 §A.1. Basic notions 287 Exercises 292 Bibliography 293 Index 297 Preface Enumerative combinatorics has seen an explosive growth over the last 50 years. The purpose of this text is to give a gentle introduction to this exciting area of research. So, rather than trying to cover many different topics, I have chosen to give a more leisurely treatment of some of the highlights of the field. My goal has been to write the exposition so it could be read by a student at the advanced undergraduate or beginning graduate level, either as part of a course or for independent study. The reader will find it similar in tone to my book on the symmetric group. I have tried to keep the prerequisites to a minimum, assuming only basic courses in linear and abstract algebra as background. Certain recurring themes are emphasized, for example, the existence of sum and prod- uct rules first for sets, then for ordinary generating functions, and finally inthecaseof exponential generating functions. I have also included some recent material from the research literature which, to my knowledge, has not appeared in book form previously, such as the theory of quotient posets and the connection between pattern avoidance and quasisymmetric functions. Most of the exercises should be doable with a reasonable amount of effort. A few unsolved conjectures have been included among the problems in the hope that an in- terested student might wish to tackle one of them. They are, of course, marked as such. A few words about the title are in order. It is in part meant to be a tip of the hat to Donald Knuth’s influential series of books The art of computer programing, Volumes 1–3 [51–53], which, among many other things, helped give birth to the study of pattern avoidance through its connection with stack sorting; see Exercise 36 in Chapter 1. I hope that the title also conveys some of the beauty found in this area of mathemat- ics, for example, the elegance of the Hook Formula (equation (7.10)) for the number of standard Young tableaux. In addition I should mention that, due to my own pref- erences, this book concentrates on the enumerative side of combinatorics and mostly ignores the important extremal and existential parts of the field. The reader interested in these areas can consult the books of Flajolet and Sedgewick [25] and of van Lint [95]. xi xii Preface This book grew out of the lecture notes which I have compiled over years of teach- ing the graduate combinatorics course at Michigan State University. I would like to thank the students in these classes for all the feedback they have given me about the various topics and their presentation. I am also indebted to the following colleagues, some of whom taught from a preliminary version of this book, who provided me with suggestions as well as catching numerous typographical errors: Matthias Beck, Moussa Benoumhani, Andreas Blass, Seth Chaiken, Sylvie Corteel, Georges Grekos, Richard Hensh, Nadia Lafrenière, Duncan Levear, and Tom Zaslavsky. Darij Grinberg deserves special mention for providing copious comments and corrections as well as providing a number of interesting exercises. I also received valuable feedback from four anony- mous referees. Finally, I wish to express my appreciation of Ina Mette, my editor at the American Mathematical Society. Without her gentle support and persistence, this text would never have seen the light of day. Because I typeset this document myself, all errors can be blamed on my computer. East Lansing, Michigan, 2020 List of Notation Symbol Definition Page 퐴(퐷) arc set of digraph 퐷 21 퐴(퐺) adjacency matrix of graph 퐺 60 풜(퐺) set of acyclic orientations of 퐺 103 푎(퐺) number of acyclic orientations of 퐺 103 퐴([푛], 푘) set of permutations 휋 in 픖푛 having 푘 descents 121 퐴(푛, 푘) Eulerian number, cardinality of 퐴([푛], 푘) 121 퐴푛(푞) Eulerian polynomial 122 풜(푃) atom set of poset 푃 169 Asc 푐 ascent set of a proper coloring 푐 279 asc 푐 ascent number of a proper coloring 푐 279 Asc 휋 ascent set of permutation 휋 76 asc 휋 ascent number of permutation 휋 76 Av푛(휋) the set of permutations in 픖푛 avoiding 휋 29 훼푟 reversal of composition 훼 32 ̄훼 expansion of composition 훼 274 훼(퐶) rank composition of chain 퐶 275 퐵(퐺) incidence matrix
Recommended publications
  • The Twelvefold Way, the Nonintersecting Circles Problem, and Partitions of Multisets
    Turkish Journal of Mathematics Turk J Math (2019) 43: 765 – 782 http://journals.tubitak.gov.tr/math/ © TÜBİTAK Research Article doi:10.3906/mat-1805-72 The twelvefold way, the nonintersecting circles problem, and partitions of multisets Toufik MANSOUR1,, Madjid MIRZVAZIRI2, Daniel YAQUBI2;∗ 1Department of Mathematics, University of Haifa, Haifa, Israel 2Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran Received: 14.05.2018 • Accepted/Published Online: 30.01.2019 • Final Version: 27.03.2019 Abstract: Let n be a nonnegative integer and A = fa1; : : : ; akg be a multiset with k positive integers such that a1 6 ··· 6 ak . In this paper, we give a recursive formula for partitions and distinct partitions of positive integer n with respect to a multiset A. We also consider the extension of the twelvefold way. By using this notion, we solve the nonintersecting circles problem, which asks to evaluate the number of ways to draw n nonintersecting circles in the plane regardless of their sizes. The latter also enumerates the number of unlabeled rooted trees with n + 1 vertices. Key words: Multiset, partitions and distinct partitions, twelvefold way, nonintersecting circles problem, rooted trees, Wilf partitions 1. Introduction A partition of n is a sequence λ1 > λ2 > ··· > λk of positive integers such that λ1 + λ2 + ··· + λk = n (see [2]). We write λ ` n to denote that λ is a partition of n. The nonzero integers λ in λ are called parts of λ. P k j j The number of parts of λ is the length of λ, denoted by `(λ), and λ = k>1 λk is the weight of λ.
    [Show full text]
  • BASIC DISCRETE MATHEMATICS Contents 1. Introduction 2 2. Graphs
    BASIC DISCRETE MATHEMATICS DAVID GALVIN, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NOTRE DAME Abstract. This document includes lecture notes, homework and exams from the Spring 2017 incarnation of Math 60610 | Basic Discrete Mathematics, a graduate course offered by the Department of Mathematics at the University of Notre Dame. The notes have been written in a single pass, and as such may well contain typographical (and sometimes more substantial) errors. Comments and corrections will be happily received at [email protected]. Contents 1. Introduction 2 2. Graphs and trees | basic definitions and questions 3 3. The extremal question for trees, and some basic properties 5 4. The enumerative question for trees | Cayley's formula 6 5. Proof of Cayley's formula 7 6. Pr¨ufer's proof of Cayley 12 7. Otter's formula 15 8. Some problems 15 9. Some basic counting problems 18 10. Subsets of a set 19 11. Binomial coefficient identities 20 12. Some problems 25 13. Multisets, weak compositions, compositions 30 14. Set partitions 32 15. Some problems 37 16. Inclusion-exclusion 39 17. Some problems 45 18. Partitions of an integer 47 19. Some problems 49 20. The Twelvefold way 49 21. Generating functions 51 22. Some problems 59 23. Operations on power series 61 24. The Catalan numbers 62 25. Some problems 69 26. Some examples of two-variable generating functions 70 27. Binomial coefficients 71 28. Delannoy numbers 71 29. Some problems 74 30. Stirling numbers of the second kind 76 Date: Spring 2017; version of December 13, 2017. 1 2 DAVID GALVIN, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NOTRE DAME 31.
    [Show full text]
  • Combinatorics—
    The Ohio State University October 10, 2016 Combinatorics | 1 Functions and sets Enumerative combinatorics (or counting), at its heart, is all about finding functions between different sets in such a way that reveals their size. A little combinatorics goes a long way: you'll probably run into it in every discipline that involves numbers. Beyond numbers and sets, there is not much additional formal theory needed to get started. For sets A and B; a function f : A ! B is any assignment of elements of B defined for every element of A: All f needs to do to be a function from A to B is that there is a rule defined for obtaining f(a) 2 B for every element of a 2 A: In some situations, it can be non-obvious that a rule in unambiguous (we will see examples), in which case it may be necessary to prove that f is well-defined. We will be interested in counting, which motivates the following three definitions. Definition 1: A function f : A ! B is injective if and only if for all x; y 2 A; f(x) = f(y) ) x = y: These are sometimes called one-to-one functions. Think of injective functions as rules for placing A inside of B in such a way that we can undo our steps. Lemma 1: A function f : A ! B is injective if and only if there is a function g : B ! A so that 8x 2 A g(f(x)) = x: This function g is called a left-inverse for f Proof.
    [Show full text]
  • Homomesies Lurking in the Twelvefold Way
    Homomesies Lurking in the Twelvefold Way Tom Roby (UConn) Describing joint research with Michael Joseph & Michael LaCroix Special Session on Enumerative Combinatorics AMS Central Sectional Meeting University of St. Thomas Minneapolis, MN USA 28 October 2016 (Friday) Slides for this talk are available online (or will be soon) at http://www.math.uconn.edu/~troby/research.html Homomesies Lurking in the Twelvefold Way Tom Roby (UConn) Describing joint research with Michael Joseph & Michael LaCroix Special Session on Enumerative Combinatorics AMS Central Sectional Meeting University of St. Thomas Minneapolis, MN USA 28 October 2016 (Friday) Slides for this talk are available online (or will be soon) at http://www.math.uconn.edu/~troby/research.html Abstract Abstract: Given a group acting on a finite set of combinatorial objects, one can often find natural statistics on these objects which are homomesic, i.e., over each orbit of the action, the average value of the statistic is the same. Since the notion was codified a few years ago, homomesic statistics have been uncovered in a wide variety of situations within dynamical algebraic combinatorics. We discuss several examples lurking in Rota's Twelvefold Way related to actions on injections, surjections (joint work with Michael Joseph), and bijections/permutations (joint work with Michael LaCroix) of finite sets. Acknowledgments This seminar talk discusses joint work with Michael Joseph and Michael La Croix. Thanks to James Propp for suggesting the study of whirling and of Foatic actions, as well as earlier collaborations on the homomesy phenomenon. Please feel free to interrupt with questions or comments. Outline Actions, orbits, and homomesy; The Twelvefold Way; Foatic actions on Sn.
    [Show full text]
  • A Species Approach to the Twelvefold
    A species approach to Rota’s twelvefold way Anders Claesson 7 September 2019 Abstract An introduction to Joyal’s theory of combinatorial species is given and through it an alternative view of Rota’s twelvefold way emerges. 1. Introduction In how many ways can n balls be distributed into k urns? If there are no restrictions given, then each of the n balls can be freely placed into any of the k urns and so the answer is clearly kn. But what if we think of the balls, or the urns, as being identical rather than distinct? What if we have to place at least one ball, or can place at most one ball, into each urn? The twelve cases resulting from exhaustively considering these options are collectively referred to as the twelvefold way, an account of which can be found in Section 1.4 of Richard Stanley’s [8] excellent Enumerative Combinatorics, Volume 1. He attributes the idea of the twelvefold way to Gian-Carlo Rota and its name to Joel Spencer. Stanley presents the twelvefold way in terms of counting functions, f : U V, between two finite sets. If we think of U as a set of balls and V as a set of urns, then requiring that each urn contains at least one ball is the same as requiring that f!is surjective, and requiring that each urn contain at most one ball is the same as requiring that f is injective. To say that the balls are identical, or that the urns are identical, is to consider the equality of such functions up to a permutation of the elements of U, or up to a permutation of the elements of V.
    [Show full text]
  • Richard Stanley's Twelvefold Way
    Richard Stanley's Twelvefold Way August 31, 2009 Many combinatorial problems can be framed as counting the number of ways to allocate balls to urns, subject to various conditions. Richard Stanley invented the \twelvefold way" to organize these results into a table with twelve entries. See his book Enumerative Combinatorics, Volume 1. Let b represent the number of balls available and u the number of urns. The following table gives the number of ways to partition the balls among the urns according to the various states of labeled or unlabeled and subject to certain restrictions. The column headed \≤ 1" corresponds to requiring that there be no more than one ball in each urn. Similarly, the column headed \≥ 1" corresponds to requiring at least one ball in each urn. Balls Urns unrestricted ≤ 1 ≥ 1 b labeled labeled u (u)b u!S(b; u) unlabeled labeled u u u b b b-u labeled unlabeled u i=1 S(b; i) [b ≤ u] S(b; u) unlabeled unlabeled u P i=1 pi(b) [b ≤ u] pu(b) P For convenient cross referencing, we will refer to each of the cases by three symbols. The rst character is an l or a u depending on whether the balls 1 are labeled or unlabeled. The second character similarly indicates whether the urns are labeled or unlabeled. The nal character is one the regular expression symbols *, ?, or + indicating no restrictions, at most one ball per urn, and at least one ball per urn respectively. 1 Labeled balls, labeled urns, unrestricted (ll*) This is the number of b-tuples of u things.
    [Show full text]
  • The Twelvefold Way
    The Twelvefold Way Lukas Bulwahn February 23, 2021 Abstract This entry provides all cardinality theorems of the Twelvefold Way. The Twelvefold Way [1, 5, 6] systematically classifies twelve related combinatorial problems concerning two finite sets, which include count- ing permutations, combinations, multisets, set partitions and number partitions. This development builds upon the existing formal develop- ments [2, 3, 4] with cardinality theorems for those structures. It pro- vides twelve bijections from the various structures to different equiva- lence classes on finite functions, and hence, proves cardinality formulae for these equivalence classes on finite functions. Contents 1 Preliminaries 4 1.1 Additions to Finite Set Theory ................. 4 1.2 Additions to Equiv Relation Theory .............. 4 1.2.1 Counting Sets by Splitting into Equivalence Classes . 5 1.3 Additions to FuncSet Theory .................. 5 1.4 Additions to Permutations Theory ............... 6 1.5 Additions to List Theory ..................... 6 1.6 Additions to Disjoint Set Theory ................ 7 1.7 Additions to Multiset Theory .................. 7 1.8 Additions to Number Partitions Theory ............ 8 1.9 Cardinality Theorems with Iverson Function .......... 8 2 Main Observations on Operations and Permutations 9 2.1 Range Multiset .......................... 9 2.1.1 Existence of a Suitable Finite Function ........ 9 2.1.2 Existence of Permutation ................ 9 2.2 Domain Partition ......................... 9 2.2.1 Existence of a Suitable Finite Function ........ 9 2.2.2 Equality under Permutation Application ........ 10 2.2.3 Existence of Permutation ................ 10 2.3 Number Partition of Range ................... 11 1 2.3.1 Existence of a Suitable Finite Function ........ 11 2.3.2 Equality under Permutation Application .......
    [Show full text]
  • Discrete Structures I – Combinatorics
    Discrete Structures I – Combinatorics Lectures by Prof. Dr. Stefan Felsner Notes taken by Elisa Haubenreißer Summer term 2011 2 Contents 1 Introductory Examples 5 1.1 Derangements..................................... .. 5 1.2 TheOfficersProblem.................................. 9 2 Basic Counting 13 n 2.1 Models for k ...................................... 14 2.2 Extending binomial coefficients . ... 15 3 Fibonacci Numbers 17 3.1 Models for F -numbers .................................. 18 3.2 FormulasforFibonacciNumbers . .... 19 3.3 The -numberSystem.................................. 21 F 3.4 ContinuedFractionsandContinuants . ..... 22 4 The Twelvefold Way 25 5 Formal Power Series 33 5.1 Bernoullinumbersandsummation . ... 34 5.2 Compositionofseries................................. .. 35 5.3 RootsofFPS ....................................... 36 5.4 CatalanNumbers.................................... 37 6 Solvinglinearrecurrencesviageneratingfunctions 39 6.1 How to compute spanning trees of Gn ......................... 40 6.2 An example with exponential generating function . ....... 42 7 q-Enumeration 43 7.1 Another model for the q-binomial............................ 47 8 Finite Sets And Posets 49 8.1 Posets–Partiallyorderedsets. ..... 50 8.2 Thediagramofaposet ................................ 51 8.3 k-intersecting families . 52 8.4 Shadows – Another prooffor Sperner’s theorem . ......... 53 8.5 Erd¨os-Ko-RadofromKruskal-Katona. ........ 55 8.6 Symmetric chain decompositions . ... 57 8.7 An application: Dedekinds problem . ... 60 9 Duality
    [Show full text]
  • Lecture Notes Combinatorics
    Lecture Notes Combinatorics Lecture by Torsten Ueckerdt (KIT) Problem Classes by Jonathan Rollin (KIT) Lecture Notes by Stefan Walzer (TU Ilmenau) Last updated: July 19, 2016 1 Contents 0 What is Combinatorics?4 1 Permutations and Combinations 10 1.1 Basic Counting Principles...................... 10 1.1.1 Addition Principle...................... 10 1.1.2 Multiplication Principle................... 10 1.1.3 Subtraction Principle.................... 11 1.1.4 Bijection Principle...................... 11 1.1.5 Pigeonhole Principle..................... 11 1.1.6 Double counting....................... 12 1.2 Ordered Arrangements { Strings, Maps and Products...... 12 1.2.1 Permutations......................... 13 1.3 Unordered Arrangements { Combinations, Subsets and Multisets................ 14 1.4 Multinomial Coefficients....................... 16 1.5 The Twelvefold Way { Balls in Boxes............... 22 1.5.1 U L: n Unlabeled Balls in k Labeled Boxes...... 22 1.5.2 L ! U: n Labeled Balls in k Unlabeled Boxes...... 24 1.5.3 L ! L: n Labeled Balls in k Labeled Boxes....... 26 1.5.4 U ! U: n Unlabeled Balls in k Unlabeled Boxes.... 28 1.5.5 Summary:! The Twelvefold Way............... 29 1.6 Binomial Coefficients { Examples and Identities.......... 30 1.7 Permutations of Sets......................... 34 1.7.1 Cycle Decompositions.................... 35 1.7.2 Transpositions........................ 38 1.7.3 Derangements......................... 40 2 Inclusion-Exclusion-Principle and M¨obiusInversion 44 2.1 The Inclusion-Exclusion Principle.................. 44 2.1.1 Applications......................... 46 2.1.2 Stronger Version of PIE................... 51 2.2 M¨obiusInversion Formula...................... 52 3 Generating Functions 57 3.1 Newton's Binomial Theorem..................... 61 3.2 Exponential Generating Functions................. 62 3.3 Recurrence Relations........................
    [Show full text]
  • Let's Expand Rota's Twelvefold Way for Counting Partitions!
    (Submitted to the American Mathematical Monthly.) Let's Expand Rota's Twelvefold Way For Counting Partitions! Robert A. Proctor Department of Mathematics University of North Carolina Chapel Hill, North Carolina 27599 rap -at- email.unc.edu January 5, 2007 Dedicated to the memory of Gian-Carlo Rota. 1. Introduction Let's suppose we have a diamond, an emerald, and a ruby. How many ways are there to store these gems in small velvet bags? The bags are identical, and we can use any number of them. Unused bags are ignored. Since the gems can move around within the bags, we may as well list the gems within each bag in alphabetical order. Since the bags are identical, we may as well record the bags in alphabetical order, according to their first elements. Using inner {}'s for the bags and outer {}'s to collect the bags, here are the possibilities: { {D}, {E}, {R} }, { {D,E}, {R} }, { {D,R}, {E} }, { {D}, {E,R} }, { {D,E,R} }. So there are 5 storage configurations. These are the "partitions" of the set {D, E, R}. For m ≥ 0, let B(m) denote the number of partitions of a set of m distinguishable items. So B(3) = 5. The counts B(0), B(1), B(2), B(3), … are called the "Bell numbers". Surprisingly, these numbers also arise in a Maclaurin series expansion: ex–1 xm e = ∑m≥0B(m)——— . m! There are dozens of variations of this notion of partition. If autographed baseballs were stored in socks, the order in which the baseballs were shoved into each sock would matter.
    [Show full text]
  • Lecture #17: Distributions (Tucker Section 5.4) David Mix Barrington 21 October 2016 Distributions
    CMPSCI 575/MATH 513 Combinatorics and Graph Theory Lecture #17: Distributions (Tucker Section 5.4) David Mix Barrington 21 October 2016 Distributions • Objects in Boxes • Unlimited Repetition: All Functions • No Repetition: One-to-One Functions • Restricted Repetition • Counting Onto Functions • The Twelvefold Way Objects in Boxes • Today we’re going to look at a new framework to describe our basic counting problems, that of distributions. • The general picture is that we have r objects that we are going to place into n boxes. We are thus choosing a function f: O → B, where O has size r and B has size n. • But there are multiple versions of the problem of counting these functions, depending on our definitions. Objects in Boxes • Suppose the objects in O are labeled with the numbers 1,…,r. (They are distinct.) Then if we put object 1 in box f(1), 2 in f(2), and so forth, our function is defined by the sequence of numbers f(1)f(2)…f(r). • But if the objects are identical, then each of our functions is characterized by just the number of objects we put in each box. Examples of Objects in Boxes • Suppose we have 100 diplomats who must each be assigned to one of five continents. If the diplomats are distinct, there are 5100 ways to do this. • If we further insist that there be 20 diplomats assigned to each continent, we can consider that the 100 diplomats are being mapped into 100 positions by a bijection, in one of 100! ways, and then correct for the over counting to get 100!/ (20!)5, which Tucker also writes as P(100; 20, 20, 20, 20, 20).
    [Show full text]
  • Combinatorics: the Art of Counting
    Prepublication copy provided to Dr Bruce Sagan. Please give confirmation to AMS by September 21, 2020. Combinatorics: The Art of Counting Not for print or electronic distribution. This file may not be posted electronically. Prepublication copy provided to Dr Bruce Sagan. Please give confirmation to AMS by September 21, 2020. Not for print or electronic distribution. This file may not be posted electronically. Prepublication copy provided to Dr Bruce Sagan. Please give confirmation to AMS by September 21, 2020. GRADUATE STUDIES IN MATHEMATICS 210 Combinatorics: The Art of Counting Bruce E. Sagan Not for print or electronic distribution. This file may not be posted electronically. Prepublication copy provided to Dr Bruce Sagan. Please give confirmation to AMS by September 21, 2020. Marco Gualtieri Bjorn Poonen Gigliola Staffilani (Chair) Jeff A. Viaclovsky 2020 Mathematics Subject Classification. Primary 05-01; Secondary 06-01. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-210 Library of Congress Cataloging-in-Publication Data Names: Sagan, Bruce Eli, 1954- author. Title: Combinatorics : the art of counting / Bruce E. Sagan. Description: Providence, Rhode Island : American Mathematical Society, [2020] | Series: Gradu- ate studies in mathematics, 1065-7339 ; 210 | Includes bibliographical references and index. | Identifiers: LCCN 2020025345 | ISBN 9781470460327 (paperback) | ISBN 9781470462802 (ebook) Subjects: LCSH: Combinatorial analysis–Textbooks. | AMS: Combinatorics – Instructional ex- position (textbooks, tutorial papers, etc.). | Order, lattices, ordered algebraic structures – Instructional exposition (textbooks, tutorial papers, etc.). Classification: LCC QA164 .S24 2020 | DDC 511/.6–dc23 LC record available at https://lccn.loc.gov/2020025345 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research.
    [Show full text]